Let f(x) = int 1 x √1 + t^2 dt
Use your calculator to find f’(1)
Sorry, it looks a little confusing now that I've written it out.
It should be an integral with an upper bound of 1 and a lower of x of √(1+t^2)
If you mean
f(x) = ∫[x,1] √(1+t^2) dt
then
f'(x) = -√(1+x^2)
so, f'(1) = -√2
This is just the chain rule in reverse. If
F(t) = ∫f(t) dt
then
F'(t) = f(t)
So, if u,v are functions of x, then
∫[u,v] f(t) dt = F(t) [u,v] = F(v)-F(u)
Now take derivatives with the chain rule, and you get
d/dx (∫[u,v] f(t) dt) = f(v) dv/dx - f(u) du/dx
To find the derivative of f(x) = ∫[1,x] √(1 + t^2) dt, we can use the fundamental theorem of calculus. However, we need a calculator or software that can perform symbolic differentiation or numerical approximation.
If you have access to a calculator or software that can perform symbolic differentiation, you can simply differentiate the integral expression. Here's how you can do it using a symbolic calculator like Wolfram Alpha:
1. Go to the Wolfram Alpha website or open the Wolfram Alpha app.
2. Type "Differentiate ∫ sqrt(1 + t^2) dt from 1 to x" (without quotes) in the input box.
3. Press Enter or submit the query.
Wolfram Alpha will calculate the derivative of the integral expression with respect to x. Once you have the derivative, you can evaluate it at x = 1 to find f'(1).
If you don't have access to a calculator or software that can perform symbolic differentiation, you can use numerical methods to approximate the derivative. Here's how you can do it using a numerical calculator or software:
1. Choose a small value for h (e.g., h = 0.01).
2. Calculate f(1 + h) and f(1), where f(x) is the integral expression.
3. Use the formula f'(1) ≈ (f(1 + h) - f(1)) / h to estimate the derivative.
4. Substitute the values you obtained into the formula to calculate the approximation of f'(1).
Note that this numerical approximation method will give you an estimate of the derivative, but it may not be as accurate as symbolic differentiation.